In this section, we
prove that the parametric representation of the simplex integral (REF ) satisfies the conformal Ward identities for any arbitrary function of cross-ratios.
The corresponding result for the vectorial simplex integral () was established in [1]}, [2]}.
Working purely in momentum space,
our approach is
to show that the action of the Ward identities on the simplex integral reduces to a total derivative. With a degree of hindsight, the structure of this total derivative, obtained in (),
can also be understood from somewhat simpler position-space arguments. We will return to these in section REF .
